Portions of this patent application contain materials that are subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever.
1. Technical Field of the Invention
This invention generally relates to a data analysis method, apparatus and article of manufacture and more particularly to apparatus, article of manufacture and analysis method for analyzing three dimensional data varying with respect to a forth independent dimension such as three dimensional time varying data.
Although the present invention finds utility in processing time varying three dimensional data, it is to be understood that any varying n dimensional data (where nxe2x89xa73) representative of a real world phenomenon such as data representative of a physical process including electrical, mechanical, biological, chemical, optical, geophysical or other process(es) may be analyzed and thereby more fully understood by applying the invention thereto. The real world n-dimensional data to which the invention finds utility include a wide variety of real world phenomena such as the behavior of a stock market, population growth, traffic flow, etc. Furthermore, the term xe2x80x9creal world n-dimensional dataxe2x80x9d also includes xe2x80x9cphysical dataxe2x80x9d representative of physical processes such as the electrical, mechanical, biological, chemical, optical, geophysical process(es) mentioned above.
Although the invention is not limited to a particular type of signal processing and includes the full range of real world data representative of processes or phenomena or combinations thereof, it is most useful when such real world n-dimensional data are nonlinear and non-stationary.
2. Description of Related Art
In the parent application, several examples of data from geophysical data signals representative of earthquakes, ocean waves, tsunamis, ocean surface elevation and wind were processed to show the invention""s wide utility to a broad variety of signal and data types. The techniques disclosed therein and elaborated upon herein represent major advances in physical data processing.
Previously, analyzing data, particularly those having nonlinear and/or nonstationary properties, was a difficult problem confronting many industries. These industries have harnessed various computer implemented methods to process data measured or otherwise taken from various processes such as electrical, mechanical, optical, biological, and chemical processes. Unfortunately, previous methods have not yielded results which are physically meaningful.
Among the difficulties found in conventional systems is that representing physical processes with physical signals may present one or more of the following problems:
(a) The total data span is too short;
(b) The data are nonstationary; and
(c) The data represent nonlinear processes.
Although problems (a)-(c) are separate issues, the first two problems are related because a data section shorter than the longest time scale of a stationary process can appear to be nonstationary. Because many physical events are transient, the data representative of those events are nonstationary. For example, a transient event such as an earthquake will produce nonstationary data when measured. Nevertheless, the nonstationary character of such data is ignored or the effects assumed to be negligible. This assumption may lead to inaccurate results and incorrect interpretation of the underlying physics as explained below.
A variety of techniques have been applied to nonlinear, nonstationary physical signals. For example, many computer implemented methods apply Fourier spectral analysis to examine the energy-frequency distribution of such signals.
Although the Fourier transform that is applied by these computer implemented methods is valid under extremely general conditions, there are some crucial restrictions: the system must be linear, and the data must be strictly periodic or stationary. If these conditions are not met, then the resulting spectrum will not make sense physically.
A common technique for meeting the linearity condition is to approximate the physical phenomena with at least one linear system. Although linear approximation is an adequate solution for some applications, many physical phenomena are highly nonlinear and do not admit a reasonably accurate linear approximation.
Furthermore, imperfect probes/sensors and numerical schemes may contaminate data representative of the phenomenon. For example, the interactions of imperfect probes with a perfect linear system can make the final data nonlinear.
Many recorded physical signals are of finite duration, nonstationary, and nonlinear because they are derived from physical processes that are nonlinear either intrinsically or through interactions with imperfect probes or numerical schemes. Under these conditions, computer implemented methods which apply Fourier spectral analysis are of limited use. For lack of alternatives, however, such methods still apply Fourier spectral analysis to process such data.
In summary, the indiscriminate use of Fourier spectral analysis in these methods and the adoption of the stationarity and linearity assumptions may give inaccurate results some of which are described below.
First, the Fourier spectrum defines uniform harmonic components globally. Therefore, the Fourier spectrum needs many additional harmonic components to simulate nonstationary data that are nonuniform globally. As a result, energy is spread over a wide frequency range.
For example, using a delta function to represent the flash of light from a lightning bolt will give a phase-locked wide white Fourier spectrum. Here, many Fourier components are added to simulate the nonstationary nature of the data in the time domain, but their existence diverts energy to a much wider frequency domain. Constrained by the conservation of energy principle, these spurious harmonics and the wide frequency spectrum cannot faithfully represent the true energy density of the lighting in the frequency and time space.
More seriously, the Fourier representation also requires the existence of negative light intensity so that the components can cancel out one another to give the final delta function representing the lightning. Thus, the Fourier components might make mathematical sense, but they often do not make physical sense when applied.
Although no physical process can be represented exactly by a delta function, some physical data such as the near field strong earthquake energy signals are of extremely short duration. Such earthquake energy signals almost approach a delta function, and they always give artificially wide Fourier spectra.
Second, Fourier spectral analysis uses a linear superposition of trigonometric functions to represent the data. Therefore, additional harmonic components are required to simulate deformed wave profiles. Such deformations, as will be shown later, are the direct consequence of nonlinear effects. Whenever the form of the data deviates from a pure sine or cosine function, the Fourier spectrum will contain harmonics.
Furthermore, both nonstationarity and nonlinearity can induce spurious harmonic components that cause unwanted energy spreading and artificial frequency smearing in the Fourier spectrum. The consequence is incorrect interpretation of physical phenomena due to the misleading energy-frequency distribution for nonlinear and nonstationary data representing the physical phenomenon.
According to the above background, the state of the art does not provide a useful computer implemented tool for analyzing nonlinear, nonstationary physical signals. Geophysical signals provide a good example of a class of signals in which this invention is applicable. Great grandparent application Ser. No. 08/872,586 filed on Jun. 10, 1997, now issued as U.S. Pat. No. 5,983,162 illustrates several types of nonlinear, nonstationary geophysical signals which are very difficult to analyze with traditional computer implemented techniques including earthquake signals, water wave signals, tsunami signals, ocean altitude and ocean circulation signals.
This application extends the technique of the parent, grandparent and great-grandparent applications to the processing of three-dimensional data and signals representative thereof. Three-dimensional data representing, for example, time varying physical phenomena are an increasing subject of various processing techniques. In fact, the above-described prior art techniques such as Fourier analysis are routinely applied to process such three-dimensional data.
Three-dimensional physical phenomena often are nonlinear and/or nonstationary. Therefore, like the one-dimensional and two-dimensional data processing techniques described above, the conventional processing techniques are simply inadequate to process such three-dimensional data.
Moreover, conventional data analysis methods are utilized to separate the various scales contained in the three-dimensional data. For example, in antarctic ice flow and antarctic ice protuberance evaluation processing, conventional spectral analysis was applied toward an objective analysis of the information contents. However, where the data contained spatial and temporal components, the presence or absence of spatial propagation was not determinable and led to erroneous results.
It is a purpose of the present invention is to solve the above-mentioned problems in conventional signal analysis techniques;
It is another purpose of the present invention to analyze three-dimensional phenomena varying with respect to a fourth independent dimension;
It is yet another purpose of the present invention to perform spectral analysis on non-stationary three-dimensional geophysical phenomenon data, such as a time sequence of gridded geophysical data maps of a particular geographical location.
The present invention permits valid intercomparison of various data sequences on an intrinsic mode-by-mode basis to enhance greatly the ability to establish teleconnections between various geophysical variables. The present invention is an apparatus, article of manufacture and method of analysis wherein a physical phenomenon is analyzed by: (1) removing a linear trend line to keep subsequent matrix operations within machine capabilities; (2) passing the data through an Hilbert transform to convert the data into complex form; (3) removing a spatial variable by producing a covariance matrix in time; (4) producing the temporal parts of the principal components by applying Singular Value Decomposition (SVD); (5) based on the rapidity with which the eigenvalues decay, select the first 3-10 principal components (PCs) for Empirical Mode Decomposition (EMD) into intrinsic modes; (6) selecting the intrinsic modes produced in order to produce either a high-pass, low-pass, or bandpass filter; (7) reconstructing a spatial part of the PCs (multiplied by the appropriate eigenvalues) by multiplying the unfiltered temporal part into the original data; and, as a final check, (8) reconstructing a filtered time series from the first 3-10 filtered complex PCs.
Computer implemented Empirical Mode Decomposition method decomposes principal components representative of a physical phenomenon into Intrinsic Mode Functions (IMFs) that are indicative of intrinsic oscillatory modes in the physical phenomenon.
Contrary to almost all the previous methods, this new computer implemented method is intuitive, direct, a posteriori, and adaptive, with the basis of the decomposition based on and derived from the physical signal. The bases so derived have no close analytic expressions, and they can only be numerically approximated in a specially programmed computer by utilizing the inventive methods disclosed herein.
More specifically, using the general method of the present invention the physical signal is analyzed without suffering the problems associated with computer implemented Fourier analysis. Namely, the present invention avoids inaccurate interpretation of the underlying physics caused in part by energy spreading and frequency smearing in the Fourier spectrum.
Further scope of applicability of the present invention will become apparent from the detailed description given hereinafter. However, it should be understood that the detailed description and specific examples, while indicating preferred embodiments of the invention, are given by way of illustration only, since various changes and modifications within the spirit and scope of the invention will become apparent to those skilled in the art from this detailed description. Furthermore, all the mathematic expressions are used as a short hand to express the inventive ideas clearly and are not limitative of the claimed invention.